His personal details and motive for defection are unknown and his exact medical condition is unclear.
The hospital will brief media on the soldier's condition at 0630 GMT, she said, without providing details. But the Ajou University Medical Center near Seoul said the soldier was relying on a breathing machine after the surgery removed the bullets.
Four North Korean soldiers chased after him, firing at least 40 rounds, including some with an AK-47 assault rifle, officials said Tuesday after reviewing security camera footage from the border.
On Monday, he first drove a military jeep but left the vehicle when one of its wheels fell into a ditch.
The United States has said that all options, including military, are on the table to deal with North Korea, although its preference is for a diplomatic solution. "Trump, during his visit, laid bare his true nature as destroyer of the world peace and stability and begged for a nuclear war on the Korean peninsula", a North Korean official said in a statement on Saturday.
The military drills come amid U.S. President Donald Trump's visit to Asia, which has been dominated by discussions over the North Korean nuclear threat.
"There's no big message" intended for North Korea or anyone else, he told reporters in an impromptu exchange in a Pentagon hallway.
He didn't elaborate on what "the last" might be, but North Korea has launched ballistic missiles that have the potential to strike the USA mainland, and it recently conducted its largest-ever underground nuclear explosion.
About 30,000 North Koreans have fled to South Korea, mostly via China, since the end of the 1950-53 Korean War.
Three U.S. aircraft carriers steam in formation in the East Sea, leading South Korean and U.S. warships during the allies' joint naval exercise on November 12, 2017, in this photo provided by South Korea's Navy.
Panmunjom, once an obscure farming village inside the 2 1/2-mile-wide Demilitarized Zone that separates the two Koreas, is where an armistice was signed to pause the Korean War. Jointly controlled by the American-led U.N. Command and North Korea, the DMZ is guarded on both sides by hundreds of thousands of combat-ready troops, razor-wire fences and tank traps.
He was shot by a North Korean soldier but succeeded in crossing, the JCS added, and has been evacuated for medical treatment.
Rex Tillerson, the secretary of state, was photographed by North Korean soldiers while he stood inside one of the huts during a visit to the DMZ in March.
It said an investigation into Monday's event was ongoing.
The Joint Security Area was the site of some bloodshed during the Cold War but there hasn't been major violence there in recent years.